FLUID MODEL FOR A DATA NETWORK WITH ALPHA FAIR BANDWIDTH
SHARING
AND GENERAL DOCUMENT SIZE DISTRIBUTIONS: TWO EXAMPLES OF STABILITY
H. C. Gromoll and R. J. Williams
Abstract
The design and analysis of congestion control mechanisms
for modern data networks
such as the Internet is a challenging problem.
Mathematical models at various levels have been introduced
in an effort to provide insight to some aspects of this problem.
A model introduced and studied by Roberts and Massoulie
aims to capture the dynamics of document
arrivals and departures in a network
where bandwidth is shared fairly amongst
flows which
correspond to continuous
transfers of individual elastic documents.
With generally distributed
interarrival times and document sizes, except for a few special
cases, it is an open problem
to establish stability of this stochastic flow level
model under the nominal condition
that the average load placed on each resource is less than its
capacity.
As a first step towards the study of this model,
in a separate work, we introduced
a measure valued process to describe the dynamic
evolution
of the residual document sizes
and proved a fluid limit result:
under mild assumptions,
rescaled measure valued processes corresponding
to a sequence of connection level models (with fixed network structure) are tight,
and any weak limit point of the sequence is almost surely a solution
of a certain fluid
model.
The invariant states for the fluid model were also characterized in
our prior work.
In this paper, we review the structure of the
stochastic flow level model, describe our fluid model approximation and then
give two interesting examples of network topologies for which
stability of the fluid model can be established
under a nominal condition.
The two types of networks are linear networks
and tree networks. The result for tree networks is
particularly interesting
as there the distribution of the number of documents
process in steady state is expected to be sensitive to the
(non-exponential) document size distribution.
Future work will be aimed at further analysis of the fluid model
and on using it for studying stability and
heavy
traffic behavior of the stochastic flow level model.
For a copy of a preprint
for personal scientific non-commercial use only, click
here for a pdf file
or
here for postscript.
This paper appears in the IMS
Collections Volume --- Markov Processes and Related Topics:
A Festschrift
for Thomas G. Kurtz, Vol. 4 (2008), 253--265.
Last updated April 27, 2007.